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o2_model [2021/04/10 06:05]
Scott Lawrence created 		o2_model [2022/12/21 04:46] (current)
Rajeev S. Erramilli [Operator content]
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 ====== $O(2)$ model ====== ====== $O(2)$ model ======
  
 +{{tag>3d}}
 +
 +The $O(2)$ model is a well studied special case of the [[on_model]]. It also goes by the name "XY model".
 +
 +===== Operator content =====
 +
 +Three of the leading operators in the spectrum: $\phi$ (an $O(2)$ vector), $s$ (a singlet), and $t$ (a symmetric rank-2 tensor i.e. a charge 2 operator) have been determined precisely and rigorously. The scaling dimensions are
 +\[
 +\Delta_\phi = 0.519088(\mathbf{22}),~\Delta_s = 1.51136(\mathbf{22}),~\Delta_t = 1.23629(\mathbf{11}).
 +\]
 +Four of OPE coefficients between these leading operators have also been determined, albeit not rigorously:
 +\[
 +\lambda_{\phi\phi s} = 0.687126(27),~\lambda_{s s s} = 0.830914(32),~\lambda_{t t s} = 1.25213(14),~\lambda_{\phi\phi t} = 1.213408(65).
 +\]
 +
 +Additionally, the flavor current and stress tensor central charges have been determined nonrigorously:
 +
 +\[
 +C_J / C_J^{\text {free}}=0.904395(28),~C_T / C_T^{\text {free}}=0.944056(15).
 +\] 
 +
 +Finally, four additional scalar operators: $s'$ (the next-to-leading singlet), $t'$ (the next-to-leading charge 2 operator), and the leading charge 3 and charge 4 operators, have been determined nonrigorously:
 +
 +\[ 
 +\Delta_{s'} = 3.794(8),~\Delta_{t'} = 3.650(2),~\Delta_{3} = 2.1086(3),~\Delta_{4}=3.14(2).
 +\]
 +
 +These data are from the bootstrap, computed here: [[https://arxiv.org/pdf/1912.03324.pdf]].
 +===== Physical realizations =====
 +
 +Superfluid helium (both He-3 and H-4), apparently.
 +
 +===== External links =====
 +
 +  * [[https://en.wikipedia.org/wiki/Classical_XY_model|Wikipedia]]
Last modified: 2021/04/10 06:05
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